Certain rings whose simple singular modules are GP-injective

We prove that if R is an idempotent reflexive left Goldie ring whose simple singular left R-modules are GP-injective, then R is a finite product of simple left Goldie rings. As a byproduct of this result we are able to show that if R is semiprime, left Goldie and left weakly π-regular, then R is a finite product of simple left Goldie rings.